In predictive regressions with variables of unknown persistence, the use of extended IV (IVX) instruments leads to asymptotically valid inference. Under highly persistent regressors, the standard normal or chi-squared limiting distributions for the usual t and Wald statistics may, however, differ markedly from the actual finite-sample distributions which exhibit in particular noncentrality. Convergence to the limiting distributions is shown to occur at a rate depending on the choice of the IVX tuning parameters and can be very slow in practice. A characterization of the leading higher-order terms of the t statistic is provided for the simple regression case, which motivates finite-sample corrections. Monte Carlo simulations confirm the usefulness of the proposed methods.

This paper compares the local power of tests for a nonlinear transformation of the dependent variable in a regression model against the alternative hypothesis of a linear transformation. It is shown that the local power of the Cox test is higher than those of the extended projection test of MacKinnon, White, and Davidson, and Bera and McAleer's test. The theoretical result is supported by a Monte-Carlo experiment in testing for a regression model with a logarithmically transformed dependent variable against a linear regression model.

In this paper, we derive uniform convergence rates of nonparametric estimators for continuous time diffusion processes. In particular, we consider kernel-based estimators of the Nadaraya–Watson type, introducing a new technical device called a damping function. This device allows us to derive sharp uniform rates over an infinite interval with minimal requirements on the processes: The existence of the moment of any order is not required and the boundedness of relevant functions can be significantly relaxed. Restrictions on kernel functions are also minimal: We allow for kernels with discontinuity, unbounded support, and slowly decaying tails. Our proofs proceed by using the covering-number technique from empirical process theory and exploiting the mixing and martingale properties of the processes. We also present new results on the path-continuity property of Brownian motions and diffusion processes over an infinite time horizon. These path-continuity results, which should also be of some independent interest, are used to control discretization biases of the nonparametric estimators. The obtained convergence results are useful for non/semiparametric estimation and testing problems of diffusion processes.

This paper considers the estimation of panel data models with interactive fixed effects where the idiosyncratic errors are subject to conditional quantile restrictions. An easy-to-implement two-step estimator is proposed for the coefficients of the observed regressors. In the first step, the principal component analysis is applied to the cross-sectional averages of the regressors to estimate the latent factors. In the second step, the smoothed quantile regression is used to estimate the coefficients of the observed regressors and the factor loadings jointly. The consistency and asymptotic normality of the estimator are established under large $N,T$ asymptotics. It is found that the asymptotic distribution of the estimator suffers from asymptotic biases, and this paper shows how to correct the biases using both analytical and split-panel jackknife bias corrections. Simulation studies confirm that the proposed estimator performs well with moderate sample sizes.

This article studies a classical problem in statistical decision theory: a hypothesis test of a sharp null in the presence of a nuisance parameter. The main contribution of this article is a characterization of two finite-sample properties often deemed reasonable in this environment: admissibility and similarity. Admissibility means that a test cannot be improved uniformly over the parameter space. Similarity requires the null rejection probability to be unaffected by the nuisance parameter.

The characterization result has two parts. The first part—established by Chernozhukov, Hansen, and Jansson (2009, Econometric Theory 25, 806–818)—states that maximizing weighted average power (WAP) subject to a similarity constraint suffices to generate admissible, similar tests. The second part—hereby established—states that constrained WAP maximization is (essentially) a necessary condition for a test to be admissible and similar. The characterization result shows that choosing an admissible, similar test is tantamount to selecting a particular weight function to report weighted average power. This result applies to full vector inference with a nuisance parameter, not to subvector inference.

The article also revisits the theory of testing in the instrumental variables model. Specifically—and in light of the relevance of the weighted average power criterion in the main theoretical result—the article suggests a weight function for the structural parameters of the homoskedastic instrumental variables model, based on the priors proposed by Chamberlain (2007). The corresponding test is, by construction, admissible and similar. In addition, the test is shown to have finite- and large-sample properties comparable to those of the conditional likelihood ratio test.

This paper develops a new estimation method for nonstationary vector autoregressions (VAR's) with unknown mixtures of I(0), I(1), and I(2) components. The method does not require prior knowledge on the exact number and location of unit roots in the system. It is, therefore, applicable for VAR's with any mixture of I(0), I(1), and I(2) variables, which may be cointegrated in any form. The limit theory for the stationary component of our estimator is still normal, thereby preserving the usual VAR limit theory. Yet, the leading term of the nonstationary component of the estimator has mixed normal limit distribution and does not involve unit root distribution. Our method is an extension of the FM-VAR procedure by Phillips (1995, Econometrica 63, 1023–1078) and yields an estimator that is optimal in the sense of Phillips (1991, Econometrica 59, 283–306). Moreover, we show for a certain class of linear restrictions that the Wald tests based on the estimator are asymptotically distributed as a weighted sum of independent chi-square variates with weights between zero and one. For such restrictions, the limit distribution of the standard Wald test is nonstandard and nuisance parameter dependent. This has a direct application for Granger-causality testing in nonstationary VAR's.

We use smoothing splines to introduce prior information in nonparametric models. The type of information we consider is based on the belief that the regression curve is similar in shape to a parametric model. The resulting estimator is a convex sum of a fit to data and the parametric model, and it can be seen as shrinkage of the smoothing spline toward the parametric model. We analyze its rates of convergence and we provide some asymptotic distribution theory. Because the asymptotic distribution is intractable, we propose to carry out inference with the estimator by using the method proposed by Politis and Romano (1994, Annals of Statistics 22, 2031–2050). We also propose a data-driven technique to compute the smoothing parameters that provides asymptotically optimal estimates. Finally, we apply our results to the estimation of a model of investment behavior of the U.S. telephone industry and we present some Monte Carlo results.

This paper proves that the fully modified vector autoregression (FM-VAR) estimator has second-order bias effects when some roots are local to unity. These bias effects are shown to result in potentially severe size distortions in FM-VAR testing when the hypothesis involves near unit root variables. In addition, the paper reveals that with the FM-VAR method near unit roots become estimated as exact unit roots with convergence speed faster than the order of the sample size. Also this result implies problems for FM-VAR testing, as such “hyperconsistent” estimates give rise to degenerate limit distributions under the null hypothesis.I am grateful to Pentti Saikkonen, Jim Stock, Markku Lanne, Jukka Nyblom, and three referees for very helpful comments on earlier drafts. This paper is a part of the research program of the Research Unit on Economic Structures and Growth (RUESG) at the Department of Economics at the University of Helsinki. Financial support from the ASLA Fulbright, the Yrjö Jahnsson Foundation, and the Finnish Cultural Foundation is gratefully acknowledged. The usual disclaimer applies.

This paper by Trygve Haavelmo was never submitted to a journal. The paper has been transcribed from Haavelmo's handwritten notes for a seminar given in 1942. The notes were kept by Haavelmo and were not known to exist by anyone else until Haavelmo died in 1999 and left his scientific remains to the Department of Economics, University of Oslo. The paper was retrieved from these personal archives by Olav Bjerkholt, who transcribed it from the original handwritten manuscript and submitted it to Econometric Theory.

Asymptotic normality is established for a class of statistics which includes as special cases weighted sum of independent and identically distributed (i.i.d.) random variables, unsigned linear rank statistics, signed rank statistics, linear combination of functions of order statistics, and linear function of concomitants of order statistics. The results obtained unify as well as extend a number of known results.

In this note we show that, when the true data generating process is a stationary one around a constant term with a break, the stationarity test of Kim (2000, Journal of Econometrics 95, 97–116) against the alternative hypothesis of change of persistence rejects the null of stationarity asymptotically with probability one.I am grateful to an anonymous referee for his useful comments, which have helped to improve the content and presentation of this note. I acknowledge financial support from Ministerio de Ciencia y Tecnología, project SEC2003-09205.

The theory of optimal transportation has experienced a sharp increase in interest in many areas of economic research such as optimal matching theory and econometric identification. A particularly valuable tool, due to its convenient representation as the gradient of a convex function, has been the Brenier map: the matching obtained as the optimizer of the Monge–Kantorovich optimal transportation problem with the euclidean distance as the cost function. Despite its popularity, the statistical properties of the Brenier map have yet to be fully established, which impedes its practical use for estimation and inference. This article takes a first step in this direction by deriving a convergence rate for the simple plug-in estimator of the potential of the Brenier map via the semi-dual Monge–Kantorovich problem. Relying on classical results for the convergence of smoothed empirical processes, it is shown that this plug-in estimator converges in standard deviation to its population counterpart under the minimax rate of convergence of kernel density estimators if one of the probability measures satisfies the Poincaré inequality. Under a normalization of the potential, the result extends to convergence in the $L^2$ norm, while the Poincaré inequality is automatically satisfied. The main mathematical contribution of this article is an analysis of the second variation of the semi-dual Monge–Kantorovich problem, which is of independent interest.

This paper highlights a tension between semiparametric efficiency and bootstrap consistency in the context of a canonical semiparametric estimation problem, namely the problem of estimating the average density. It is shown that although simple plug-in estimators suffer from bias problems preventing them from achieving semiparametric efficiency under minimal smoothness conditions, the nonparametric bootstrap automatically corrects for this bias and that, as a result, these seemingly inferior estimators achieve bootstrap consistency under minimal smoothness conditions. In contrast, several “debiased” estimators that achieve semiparametric efficiency under minimal smoothness conditions do not achieve bootstrap consistency under those same conditions.

The first novelty of this paper is that we show global identification of the private values distribution in a sealed-bid third-price auction model using a fully nonparametric methodology. The second novelty of the paper comes from the study of the identification and estimation of the model using a quantile approach. We consider an i.i.d. private values environment with risk-averse bidders. In the first place, we consider the case where the risk-aversion parameter is known. We show that the speed of convergence in process of our nonparametric estimator produces at the root-n parametric rate, and we explain the intuition behind this apparently surprising result. Next, we consider that the risk-aversion parameter is unknown, and we locally identify it using exogenous variation in the number of participants. We extend our procedure to the case where we observe only the bids corresponding to the transaction prices, and we generalize the model so as to account for the presence of exogenous variables. The methodological toolbox used to analyse identification of the third-price auction model can be employed in the study of other games of incomplete information. Our results are interesting, also from a policy perspective, as some authors recommend the use of the third-price auction format for certain Internet auctions. Moreover, we contribute to the econometric literature on auctions using a quantile approach.

The finite sample performance of spectral regression estimators in temporally aggregated cointegrated systems is investigated via the use of simulation experiments. The simulations address issues such as “optimal” choice of bandwidth parameter and effects of smoothing kernel in constructing estimates of spectral densities that are used by the spectral regression estimators; the effects of stock and flow variables and mixtures of the two, including the relative finite sample efficiency of the estimators under different combinations of stock and flow variables; and the effects of conducting iterations of the spectral estimators. A striking feature of the results is the crucial role that correct choice of bandwidth and kernel function plays in producing accurate estimates of the unknown parameters. Furthermore, estimates obtained using flow data alone are found to be more efficient, in the sense of having smaller variance, than those obtained using stock data alone or mixtures of stocks and flows, thereby confirming in finite samples their relative asymptotic properties.

It was recently shown (Abadir, 1993b) that nonstationarity causes the limiting distributions of the Wald (W) and Lagrange multiplier (LM) statistics to become different from each other. This paper demonstrates that such a divergence between the two distributions can be used as an indicator of the presence of a unit root. A test based on this idea is devised by modifying the normalized autocorrelation coefficient (NAC). It is then shown to be an improvement on NAC in large samples and an improvement on other existing tests in large effective samples. The paper also investigates the effect of nonstationarity on the well-known inequality W ≥ LR ≥ LM.

When in the class of “exact” present value (PV) relations the decision variables do not Granger cause the explanatory variables, and a vector autoregressive (VAR) process is used to derive the cross-equation restrictions, the system embodies explosive roots, which hardly can be reconciled with the typical features observed in most macroeconomic time series. This paper investigates the issue.I thank Paolo Paruolo and two anonymous referees for helpful comments and suggestions on earlier drafts of the paper. I am solely responsible for all remaining errors.

Vector time series analysis has become a standard tool in macroeconometrics and in empirical finance since the early 1980s. Over the last two decades, Helmut Lütkepohl has published extensively on virtually all topics covered in this book, which puts him in an excellent position for writing an authoritative textbook on this subject.